Question

Explain geometrically how the Midpoint Rule is used to approximate a definite integral.

Short answer

Answer: The Midpoint Rule is a numerical integration method for approximating a definite integral by calculating the area of rectangles with bases on subintervals and heights equal to the function value at the midpoint of each subinterval. The step-by-step process involves: 1. Understanding the Midpoint Rule: approximating the definite integral of a function using rectangles based on function midpoints. 2. Dividing the interval into subintervals: Divide the interval [a, b] into n equal subintervals with width Δx = (b - a)/n. 3. Determining the midpoints: Calculate the midpoint m_i of each subinterval as m_i = (x_{i-1} + x_i)/2 and find the function value at these midpoints, f(m_i). 4. Calculating the rectangles' areas: Multiply the height f(m_i) by the width Δx to find the area A_i of each rectangle, A_i = f(m_i) * Δx. 5. Approximating the definite integral: Sum all the rectangles' areas, ∫[a, b] f(x)dx ≈ Σ A_i. Using more subintervals (larger n) generally leads to a more accurate approximation.

Step-by-step solution

Understanding the Midpoint Rule

To explain the Midpoint Rule, it's important to first understand what it means. The Midpoint Rule is a simple but effective method for approximating the definite integral of a function on a given interval, by calculating the area of rectangles whose heights are determined by the function value at the midpoint of each subinterval.

Dividing the interval into subintervals

To start, divide the interval of integration [a, b] into n equal subintervals with a width of Δx, where Δx = (b - a)/n. These subintervals will form the base of the rectangles used to approximate the area under the curve.

Determine the midpoint of each subinterval

For each subinterval [x_{i-1}, x_i], find its midpoint m_i by averaging the upper and lower bounds of the subinterval: m_i = (x_{i-1} + x_i)/2. The heights of the rectangles will be determined by evaluating the function at the midpoint: f(m_i).

Calculate the area of the rectangles

To find the area of each rectangle, multiply its height f(m_i) by its width Δx. The area A_i of the ith rectangle can be calculated as: A_i = f(m_i) * Δx.

Approximate the definite integral

To approximate the definite integral, sum the areas of all rectangles: ∫[a, b] f(x)dx ≈ Σ A_i. This is known as the Midpoint Rule approximation. Keep in mind that, in general, more subintervals (larger n) will lead to a more accurate approximation. Through this step-by-step process, the Midpoint Rule can be used to geometrically approximate a definite integral by calculating the sum of areas of rectangles with bases on the subintervals and heights equal to the function value at the midpoint of each subinterval.

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus that represents the area under a curve over a specific interval on the x-axis. This area can be calculated precisely using the definite integral form: \[\int_{a}^{b} f(x) \, dx\]where \(a\) and \(b\) are the bounds of the interval, and \(f(x)\) is the function defining the curve.
Understanding definite integrals is crucial because they allow us to quantify the total accumulation of values represented by \(f(x)\) from point \(a\) to \(b\).
These calculations are essential in fields such as physics for calculating quantities like displacement and area.
When the function is complicated or does not have an easy antiderivative, we often resort to approximation methods like the Midpoint Rule to estimate the definite integral.

Geometric Approximation

Geometric approximation is a method used to estimate the value of definite integrals without calculating them exactly.
This technique is particularly useful when dealing with complex functions or when a precise analytic solution is difficult to achieve. Through shapes like rectangles, we approximate the area under a curve to understand its behavior over an interval.
The Midpoint Rule is one such method and falls under geometric approximation, utilizing the simple concept of area of a rectangle to estimate the integral's value.
The idea is by summing the areas of several rectangles, the total area closely reflects the actual area under the curve, especially when the rectangles are narrow and numerous.

Rectangular Approximation

Rectangular approximation is a straightforward visualization method to estimate integrals, whereby rectangles represent small segments of the area under a curve.
In the Midpoint Rule, each rectangle's height is defined by the function's value at the midpoint of the subinterval over which the rectangle is based.
Key aspects of rectangular approximation include:

• Base width (Δx): The width of each rectangle, calculated as \(\frac{b-a}{n}\), where \(b-a\) is the interval width and \(n\) is the number of subintervals.
• Height: The height of the rectangle is given by the function’s value at the midpoint \(f(m_i)\).
• Area: The area of each rectangle is \(f(m_i) * Δx\).

It’s through these rectangles that we sum their areas to approximate the integral. The more rectangles, the better the approximation.

Subintervals

Subintervals are essential in the process of approximating a definite integral using the Midpoint Rule.
The interval \([a, b]\) is divided into \(n\) equal parts called subintervals to form the basis of rectangles in the approximation process.
The characteristics of subintervals are significant because:

• Equal length: Each subinterval has the same width, \(Δx = \frac{b-a}{n}\).
• Midpoint Calculation: For each subinterval \([x_{i-1}, x_i]\), a midpoint \(m_i\) is calculated as \(\frac{x_{i-1} + x_i}{2}\).
• Function Evaluation: The function \(f(x)\) is evaluated at this midpoint to determine the rectangle's height.

Choosing more subintervals (i.e., a larger \(n\)) can lead to a more accurate approximation, as this generally allows the rectangles to more closely fit the curve.